Open Access

Power series method and approximate linear differential equations of second order

Advances in Difference Equations20132013:76

DOI: 10.1186/1687-1847-2013-76

Received: 14 December 2012

Accepted: 4 March 2013

Published: 26 March 2013

Abstract

In this paper, we will establish a theory for the power series method that can be applied to various types of linear differential equations of second order to prove the Hyers-Ulam stability.

MSC:34A05, 39B82, 26D10, 34A40.

Keywords

power series method approximate linear differential equation simple harmonic oscillator equation Hyers-Ulam stability approximation

1 Introduction

Let X be a normed space over a scalar field K https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq1_HTML.gif, and let I R https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq2_HTML.gif be an open interval, where K https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq1_HTML.gif denotes either or . Assume that a 0 , a 1 , , a n : I K https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq3_HTML.gif and g : I X https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq4_HTML.gif are given continuous functions. If for every n times continuously differentiable function y : I X https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq5_HTML.gif satisfying the inequality
a n ( x ) y ( n ) ( x ) + a n 1 ( x ) y ( n 1 ) ( x ) + + a 1 ( x ) y ( x ) + a 0 ( x ) y ( x ) + g ( x ) ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equa_HTML.gif
for all x I https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq6_HTML.gif and for a given ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq7_HTML.gif, there exists an n times continuously differentiable solution y 0 : I X https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq8_HTML.gif of the differential equation
a n ( x ) y ( n ) ( x ) + a n 1 ( x ) y ( n 1 ) ( x ) + + a 1 ( x ) y ( x ) + a 0 ( x ) y ( x ) + g ( x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equb_HTML.gif

such that y ( x ) y 0 ( x ) K ( ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq9_HTML.gif for any x I https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq6_HTML.gif, where K ( ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq10_HTML.gif is an expression of ε with lim ε 0 K ( ε ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq11_HTML.gif, then we say that the above differential equation has the Hyers-Ulam stability. For more detailed definitions of the Hyers-Ulam stability, we refer the reader to [18].

Obłoza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [9, 10]). Thereafter, Alsina and Ger [11] proved the Hyers-Ulam stability of the differential equation y ( x ) = y ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq12_HTML.gif. It was further proved by Takahasi et al. that the Hyers-Ulam stability holds for the Banach space valued differential equation y ( x ) = λ y ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq13_HTML.gif (see [12] and also [1315]).

Moreover, Miura et al. [16] investigated the Hyers-Ulam stability of an n th-order linear differential equation. The first author also proved the Hyers-Ulam stability of various linear differential equations of first order (ref. [1725]).

Recently, the first author applied the power series method to studying the Hyers-Ulam stability of several types of linear differential equations of second order (see [2634]). However, it was inconvenient that he had to alter and apply the power series method with respect to each differential equation in order to study the Hyers-Ulam stability. Thus, it is inevitable to develop a power series method that can be comprehensively applied to different types of differential equations.

In Sections 2 and 3 of this paper, we establish a theory for the power series method that can be applied to various types of linear differential equations of second order to prove the Hyers-Ulam stability.

Throughout this paper, we assume that the linear differential equation of second order of the form
p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equ1_HTML.gif
(1)
for which x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq14_HTML.gif is an ordinary point, has the general solution y h : ( ρ 0 , ρ 0 ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq15_HTML.gif, where ρ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq16_HTML.gif is a constant with 0 < ρ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq17_HTML.gif and the coefficients p , q , r : ( ρ 0 , ρ 0 ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq18_HTML.gif are analytic at 0 and have power series expansions
p ( x ) = m = 0 p m x m , q ( x ) = m = 0 q m x m and r ( x ) = m = 0 r m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equc_HTML.gif

for all x ( ρ 0 , ρ 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq19_HTML.gif. Since x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq14_HTML.gif is an ordinary point of (1), we remark that p 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq20_HTML.gif.

2 Inhomogeneous differential equation

In the following theorem, we solve the linear inhomogeneous differential equation of second order of the form
p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = m = 0 a m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equ2_HTML.gif
(2)

under the assumption that x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq14_HTML.gif is an ordinary point of the associated homogeneous linear differential equation (1).

Theorem 2.1 Assume that the radius of convergence of power series m = 0 a m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq21_HTML.gif is ρ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq22_HTML.gif and that there exists a sequence { c m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq23_HTML.gif satisfying the recurrence relation
k = 0 m [ ( k + 2 ) ( k + 1 ) c k + 2 p m k + ( k + 1 ) c k + 1 q m k + c k r m k ] = a m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equ3_HTML.gif
(3)
for any m N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq24_HTML.gif. Let ρ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq25_HTML.gif be the radius of convergence of power series m = 0 c m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq26_HTML.gif and let ρ 3 = min { ρ 0 , ρ 1 , ρ 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq27_HTML.gif, where ( ρ 0 , ρ 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq28_HTML.gif is the domain of the general solution to (1). Then every solution y : ( ρ 3 , ρ 3 ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq29_HTML.gif of the linear inhomogeneous differential equation (2) can be expressed by
y ( x ) = y h ( x ) + m = 0 c m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equd_HTML.gif

for all x ( ρ 3 , ρ 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq30_HTML.gif, where y h ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq31_HTML.gif is a solution of the linear homogeneous differential equation (1).

Proof Since x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq14_HTML.gif is an ordinary point, we can substitute m = 0 c m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq32_HTML.gif for y ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq33_HTML.gif in (2) and use the formal multiplication of power series and consider (3) to get
p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = m = 0 k = 0 m p m k ( k + 2 ) ( k + 1 ) c k + 2 x m + m = 0 k = 0 m q m k ( k + 1 ) c k + 1 x m + m = 0 k = 0 m r m k c k x m = m = 0 k = 0 m [ ( k + 2 ) ( k + 1 ) c k + 2 p m k + ( k + 1 ) c k + 1 q m k + c k r m k ] x m = m = 0 a m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Eque_HTML.gif
for all x ( ρ 3 , ρ 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq30_HTML.gif. That is, m = 0 c m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq26_HTML.gif is a particular solution of the linear inhomogeneous differential equation (2), and hence every solution y : ( ρ 3 , ρ 3 ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq29_HTML.gif of (2) can be expressed by
y ( x ) = y h ( x ) + m = 0 c m x m , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equf_HTML.gif

where y h ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq34_HTML.gif is a solution of the linear homogeneous differential equation (1). □

For the most common case in applications, the coefficient functions p ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq35_HTML.gif, q ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq36_HTML.gif, and r ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq37_HTML.gif of the linear differential equation (1) are simple polynomials. In such a case, we have the following corollary.

Corollary 2.2 Let p ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq35_HTML.gif, q ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq36_HTML.gif, and r ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq37_HTML.gif be polynomials of degree at most d 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq38_HTML.gif. In particular, let d 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq39_HTML.gif be the degree of p ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq35_HTML.gif. Assume that the radius of convergence of power series m = 0 a m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq21_HTML.gif is ρ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq22_HTML.gif and that there exists a sequence { c m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq23_HTML.gif satisfying the recurrence formula
k = m 0 m [ ( k + 2 ) ( k + 1 ) c k + 2 p m k + ( k + 1 ) c k + 1 q m k + c k r m k ] = a m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equ4_HTML.gif
(4)
for any m N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq24_HTML.gif, where m 0 = max { 0 , m d } https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq40_HTML.gif. If the sequence { c m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq23_HTML.gif satisfies the following conditions:
  1. (i)

    lim m c m 1 / m c m = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq41_HTML.gif,

     
  2. (ii)

    there exists a complex number L such that lim m c m / c m 1 = L https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq42_HTML.gif and p d 0 + L p d 0 1 + + L d 0 1 p 1 + L d 0 p 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq43_HTML.gif,

     
then every solution y : ( ρ 3 , ρ 3 ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq29_HTML.gif of the linear inhomogeneous differential equation (2) can be expressed by
y ( x ) = y h ( x ) + m = 0 c m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equg_HTML.gif

for all x ( ρ 3 , ρ 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq30_HTML.gif, where ρ 3 = min { ρ 0 , ρ 1 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq44_HTML.gif and y h ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq31_HTML.gif is a solution of the linear homogeneous differential equation (1).

Proof Let m be any sufficiently large integer. Since p d + 1 = p d + 2 = = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq45_HTML.gif, q d + 1 = q d + 2 = = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq46_HTML.gif and r d + 1 = r d + 2 = = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq47_HTML.gif, if we substitute m d + k https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq48_HTML.gif for k in (4), then we have
a m = k = 0 d [ ( m d + k + 2 ) ( m d + k + 1 ) c m d + k + 2 p d k + ( m d + k + 1 ) c m d + k + 1 q d k + c m d + k r d k ] . https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equh_HTML.gif
By (i) and (ii), we have
lim sup m | a m | 1 / m = lim sup m | k = 0 d ( m d + k + 2 ) ( m d + k + 1 ) c m d + k + 2 × ( p d k + q d k ( m d + k + 2 ) c m d + k + 1 c m d + k + 2 + r d k ( m d + k + 2 ) ( m d + k + 1 ) c m d + k c m d + k + 1 c m d + k + 1 c m d + k + 2 ) | 1 / m = lim sup m | k = 0 d ( m d + k + 2 ) ( m d + k + 1 ) c m d + k + 2 p d k | 1 / m = lim sup m | k = d d 0 d ( m d + k + 2 ) ( m d + k + 1 ) c m d + k + 2 p d k | 1 / m = lim sup m | ( m d 0 + 2 ) ( m d 0 + 1 ) c m d 0 + 2 ( p d 0 + L p d 0 1 + + L d 0 p 0 ) | 1 / m = lim sup m | ( p d 0 + L p d 0 1 + + L d 0 p 0 ) ( m d 0 + 2 ) ( m d 0 + 1 ) | 1 / m × ( | c m d 0 + 2 | 1 / ( m d 0 + 2 ) ) ( m d 0 + 2 ) / m = lim sup m | c m d 0 + 2 | 1 / ( m d 0 + 2 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equi_HTML.gif

which implies that the radius of convergence of the power series m = 0 c m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq26_HTML.gif is ρ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq49_HTML.gif. The rest of this corollary immediately follows from Theorem 2.1. □

In many cases, it occurs that p ( x ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq50_HTML.gif in (1). For this case, we obtain the following corollary.

Corollary 2.3 Let ρ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq51_HTML.gif be a distance between the origin 0 and the closest one among singular points of q ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq52_HTML.gif, r ( z ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq53_HTML.gif, or m = 0 a m z m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq54_HTML.gif in a complex variable z. If there exists a sequence { c m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq23_HTML.gif satisfying the recurrence relation
( m + 2 ) ( m + 1 ) c m + 2 + k = 0 m [ ( k + 1 ) c k + 1 q m k + c k r m k ] = a m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equ5_HTML.gif
(5)
for any m N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq24_HTML.gif, then every solution y : ( ρ 3 , ρ 3 ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq29_HTML.gif of the linear inhomogeneous differential equation
y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = m = 0 a m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equ6_HTML.gif
(6)
can be expressed by
y ( x ) = y h ( x ) + m = 0 c m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equj_HTML.gif

for all x ( ρ 3 , ρ 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq30_HTML.gif, where y h ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq31_HTML.gif is a solution of the linear homogeneous differential equation (1) with p ( x ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq50_HTML.gif.

Proof If we put p 0 = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq55_HTML.gif and p i = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq56_HTML.gif for each i N https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq57_HTML.gif, then the recurrence relation (3) reduces to (5). As we did in the proof of Theorem 2.1, we can show that m = 0 c m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq26_HTML.gif is a particular solution of the linear inhomogeneous differential equation (6).

According to [[35], Theorem 7.4] or [[36], Theorem 5.2.1], there is a particular solution y 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq58_HTML.gif of (6) in a form of power series in x whose radius of convergence is at least ρ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq51_HTML.gif. Moreover, since m = 0 c m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq26_HTML.gif is a solution of (6), it can be expressed as a sum of both y 0 ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq58_HTML.gif and a solution of the homogeneous equation (1) with p ( x ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq50_HTML.gif. Hence, the radius of convergence of m = 0 c m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq26_HTML.gif is at least ρ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq51_HTML.gif.

Now, every solution y : ( ρ 3 , ρ 3 ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq29_HTML.gif of (6) can be expressed by
y ( x ) = y h ( x ) + m = 0 c m x m , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equk_HTML.gif

where y h ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq34_HTML.gif is a solution of the linear differential equation (1) with p ( x ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq50_HTML.gif. □

3 Approximate differential equation

In this section, let ρ 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq22_HTML.gif be a constant. We denote by C https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq59_HTML.gif the set of all functions y : ( ρ 1 , ρ 1 ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq60_HTML.gif with the following properties:
  1. (a)

    y ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq33_HTML.gif is expressible by a power series m = 0 b m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq61_HTML.gif whose radius of convergence is at least ρ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq49_HTML.gif;

     
  2. (b)
    There exists a constant K 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq62_HTML.gif such that m = 0 | a m x m | K | m = 0 a m x m | https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq63_HTML.gif for any x ( ρ 1 , ρ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq64_HTML.gif, where
    a m = k = 0 m [ ( k + 2 ) ( k + 1 ) b k + 2 p m k + ( k + 1 ) b k + 1 q m k + b k r m k ] https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equl_HTML.gif
     

for all m N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq24_HTML.gif and p 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq20_HTML.gif.

Lemma 3.1 Given a sequence { a m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq65_HTML.gif, let { c m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq23_HTML.gif be a sequence satisfying the recurrence formula (3) for all m N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq24_HTML.gif. If p 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq20_HTML.gif and n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq66_HTML.gif, then c n https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq67_HTML.gif is a linear combination of a 0 , a 1 , , a n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq68_HTML.gif, c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq69_HTML.gif, and c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq70_HTML.gif.

Proof We apply induction on n. Since p 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq20_HTML.gif, if we set m = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq71_HTML.gif in (3), then
c 2 = 1 2 p 0 a 0 r 0 2 p 0 c 0 q 0 2 p 0 c 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equm_HTML.gif
i.e., c 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq72_HTML.gif is a linear combination of a 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq73_HTML.gif, c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq69_HTML.gif, and c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq70_HTML.gif. Assume now that n is an integer not less than 2 and c i https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq74_HTML.gif is a linear combination of a 0 , , a i 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq75_HTML.gif, c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq69_HTML.gif, c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq70_HTML.gif for all i { 2 , 3 , , n } https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq76_HTML.gif, namely,
c i = α i 0 a 0 + α i 1 a 1 + + α i i 2 a i 2 + β i c 0 + γ i c 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equn_HTML.gif
where α i 0 , , α i i 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq77_HTML.gif, β i https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq78_HTML.gif, γ i https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq79_HTML.gif are complex numbers. If we replace m in (3) with n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq80_HTML.gif, then
a n 1 = 2 c 2 p n 1 + c 1 q n 1 + c 0 r n 1 + 6 c 3 p n 2 + 2 c 2 q n 2 + c 1 r n 2 + + n ( n 1 ) c n p 1 + ( n 1 ) c n 1 q 1 + c n 2 r 1 + ( n + 1 ) n c n + 1 p 0 + n c n q 0 + c n 1 r 0 = ( n + 1 ) n p 0 c n + 1 + [ n ( n 1 ) p 1 + n q 0 ] c n + + ( 2 p n 1 + 2 q n 2 + r n 3 ) c 2 + ( q n 1 + r n 2 ) c 1 + r n 1 c 0 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equo_HTML.gif
which implies
c n + 1 = 1 ( n + 1 ) n p 0 a n 1 n ( n 1 ) p 1 + n q 0 ( n + 1 ) n p 0 c n 2 p n 1 + 2 q n 2 + r n 3 ( n + 1 ) n p 0 c 2 q n 1 + r n 2 ( n + 1 ) n p 0 c 1 r n 1 ( n + 1 ) n p 0 c 0 = α n + 1 0 a 0 + α n + 1 1 a 1 + + α n + 1 n 1 a n 1 + β n + 1 c 0 + γ n + 1 c 1 , https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equp_HTML.gif

where α n + 1 0 , , α n + 1 n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq81_HTML.gif, β n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq82_HTML.gif, γ n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq83_HTML.gif are complex numbers. That is, c n + 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq84_HTML.gif is a linear combination of a 0 , a 1 , , a n 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq85_HTML.gif, c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq69_HTML.gif, c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq70_HTML.gif, which ends the proof. □

In the following theorem, we investigate a kind of Hyers-Ulam stability of the linear differential equation (1). In other words, we answer the question whether there exists an exact solution near every approximate solution of (1). Since x = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq14_HTML.gif is an ordinary point of (1), we remark that p 0 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq20_HTML.gif.

Theorem 3.2 Let { c m } https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq23_HTML.gif be a sequence of complex numbers satisfying the recurrence relation (3) for all m N 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq24_HTML.gif, where (b) is referred for the value of a m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq86_HTML.gif, and let ρ 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq25_HTML.gif be the radius of convergence of the power series m = 0 c m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq26_HTML.gif. Define ρ 3 = min { ρ 0 , ρ 1 , ρ 2 } https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq27_HTML.gif, where ( ρ 0 , ρ 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq28_HTML.gif is the domain of the general solution to (1). Assume that y : ( ρ 1 , ρ 1 ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq60_HTML.gif is an arbitrary function belonging to C https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq59_HTML.gif and satisfying the differential inequality
| p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) | ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equ7_HTML.gif
(7)
for all x ( ρ 3 , ρ 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq87_HTML.gif and for some ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq7_HTML.gif. Let α n 0 , α n 1 , , α n n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq88_HTML.gif, β n https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq89_HTML.gif, γ n https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq90_HTML.gif be the complex numbers satisfying
c n = α n 0 a 0 + α n 1 a 1 + + α n n 2 a n 2 + β n c 0 + γ n c 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equ8_HTML.gif
(8)
for any integer n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq66_HTML.gif. If there exists a constant C > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq91_HTML.gif such that
| α n 0 a 0 + α n 1 a 1 + + α n n 2 a n 2 | C | a n | https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equ9_HTML.gif
(9)
for all integers n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq66_HTML.gif, then there exists a solution y h : ( ρ 3 , ρ 3 ) C https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq92_HTML.gif of the linear homogeneous differential equation (1) such that
| y ( x ) y h ( x ) | C K ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equq_HTML.gif

for all x ( ρ 3 , ρ 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq87_HTML.gif, where K is the constant determined in (b).

Proof By the same argument presented in the proof of Theorem 2.1 with m = 0 b m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq61_HTML.gif instead of m = 0 c m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq26_HTML.gif, we have
p ( x ) y ( x ) + q ( x ) y ( x ) + r ( x ) y ( x ) = m = 0 a m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equ10_HTML.gif
(10)
for all x ( ρ 3 , ρ 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq87_HTML.gif. In view of (b), there exists a constant K 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq62_HTML.gif such that
m = 0 | a m x m | K | m = 0 a m x m | https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equ11_HTML.gif
(11)

for all x ( ρ 1 , ρ 1 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq93_HTML.gif.

Moreover, by using (7), (10), and (11), we get
m = 0 | a m x m | K | m = 0 a m x m | K ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equr_HTML.gif

for any x ( ρ 3 , ρ 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq87_HTML.gif. (That is, the radius of convergence of power series m = 0 a m x m https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq21_HTML.gif is at least ρ 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq51_HTML.gif.)

According to Theorem 2.1 and (10), y ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq33_HTML.gif can be written as
y ( x ) = y h ( x ) + n = 0 c n x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equ12_HTML.gif
(12)

for all x ( ρ 3 , ρ 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq87_HTML.gif, where y h ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq31_HTML.gif is a solution of the homogeneous differential equation (1). In view of Lemma 3.1, the c n https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq67_HTML.gif can be expressed by a linear combination of the form (8) for each integer n 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq66_HTML.gif.

Since n = 0 c n x n https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq94_HTML.gif is a particular solution of (2), if we set c 0 = c 1 = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq95_HTML.gif, then it follows from (8), (9), and (12) that
| y ( x ) y h ( x ) | n = 0 | c n x n | C K ε https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_Equs_HTML.gif

for all x ( ρ 3 , ρ 3 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-1847-2013-76/MediaObjects/13662_2012_Article_413_IEq87_HTML.gif. □

Declarations

Acknowledgements

Dedicated to Professor Hari M Srivastava.

This research was completed with the support of The Scientific and Technological Research Council of Turkey while the first author was a visiting scholar at Istanbul Commerce University, Istanbul, Turkey.

Authors’ Affiliations

(1)
Mathematics Section, College of Science and Technology, Hongik University
(2)
Department of Mathematics, Faculty of Sciences and Arts, Istanbul Commerce University

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